## Functions

## Functions in 2D and 3D

Functions of the form F(x) and F(x,y) containing up to 20 subfunctions, 30 special functions, 150 numerical and 100 named constants.

Functions of the form F(x) can be integrated, differentiated or searched for zeroes and extrema. Cartesian as well as Polar coordinates can be chosen and functions can be represented in parametrized form (i.e. y(t) versus x(t)).

F(x,y) can be represented by Shaded surfaces, Contourplots and Cross-sections through Contourplots. The 3D viewer provides a quick view of the shaded surface from different viewing angles and distances. Rotate the surface, zoom in/out or change the position of the light source by moving the mouse.

## Special Functions (real numbers only!)

## Random numbers

## Random(a)

Random(a) Generates a series of random numbers

between 0 and 1 from seed a. Below an example is given

F1=Random(0.5) calculated

for 100 x-values (F is not a function of x here)

## Random steps with probability x: Ranstep(a,x)

Generates a random series of 0’s and 1’s from seed a.

The average fraction (or probablity) of 1’s is given by x.

Below an example is given of an Iteration (100 iterations):

F1=F2

F2=F1+ Ranstep(0.5,0.3)

## Modulo function Mod(a,x)

Mod(a,x) = x – k.a, where integer k is chosen such that

0 < Mod(a,x) < a for a > 0 and

a < Mod(a,x) < 0 for a < 0

## Fresnel Integrals

## Sine and Cosine Fresnel Integral: C(x), S(x)

## Exponential integrals and related functions

## (Scaled) Exponential integral En(n,x)

Note the exponential integral is often defined without the scaling factor exp(x)

## Logarithmic integral Li(x)

## Sine integral S(x)

## Cosine integral C(x)

## Hyperbolic sine integral Shi(x)

## Hyperbolic cosine integral Chi(x)

## Gamma function and related functions

## Incomplete (lower) gamma function of the first kind Gammi(a,x)

Mathgrapher calculates the normalized function:

## Complementary (upper) incomplete gamma function of the second kind Gammi(a,x)

## Riemann Zeta function Zeta(x)

This series is convergent for all numbers s>1. This holds for all complex numbers with real part >1. Euler showed that it is connected with the prime numbers

by the relation:

where the product runs over all prime numbers p.

Riemann found an analytical continuation to this function

for all s except 1

The integral runs from infinity to zero just above the real

axis, turns aorund zero and runs back to infinity just below the real axis

Mathgrapher calculates this function along the real axis (s=1 excluded)

The famous Riemann hypothesis states that all (non-trivial) zeroes of this function are on the line Re(s)=1/2. This Riemann hypothesis is unproved although more than

10^13 zeroes have been found, all of which ly on this line.

## Bessel functions

## Bessel functions of the first kind Bessel J(a,x)

## Bessel functions of the second kind Bessel Y(a,x)

## Elliptic integrals

## Elliptic integral of the first kind K(x)

## Elliptic integral of the second kind E(x)

## Airy functions AI(x), BI(x)

The Airy functions AI(x) and BI(x) are the two linearly independant solutions of the second-order differential equation: y”-xy=0. For real values these functions are equal to the following integrals:

and

## Probability function

## Poisson probability function (or probability distribution function) Ppf(a,x)

Example:

The distribution gives the probability for m radioactive

decays in some fixed time interval, where a is the mean

number of events in that interval. For example: a = np, the product of the the number of radioactive nuclei in a source and the probability for a nucleus to decay in a that interval of time.

## Binomial probability distribution function Bpf(a,b,x)

Gives the probability of m successes in n trials when the chance of success is p. In Mathgrapher this fuction is Bpf(a,b,x) where a=n, b=p, x=m

## Distribution functions

## Binomial (cumulative) distribution function Bin(a,b,x)

Gives the probability that m or fewer successes occur in n independent trials when the chance of success is p. In Mathgrapher this is Bin(a,b,x) where a=n (integer), b=p and x=m (integer).

## Poisson (cumulative) distribution function Pdf(a,x)

Gives the probability of m or less events in some fixed time interval, where a is the mean rate of events.

## Error Function Erf(x)

## Gaussian (standard normal) distribution function Gausdf(x)

A random distribution is often described by the Gauss probability distribution function

Mathgrapher gives the standard cumulative Gauss distribution function

Below some examples are given for this function Gausdf(x)

## (Incomplete) Beta distribution function Betadf(a,b,x)

## Chi squared distribution function (cumulative) Chidf(a,x)

The denumerator contains the Gamma function. In Mathgrapher this is Chidf(a,x) where a is the number of

degrees of freedom (a= 0.5 or larger).

## Student’s t distribution function (cumulative) Tdf(a,x)

The (de)numerator contains the Gamma function. In Mathgrapher this is Tdf(a,x) where a is the number of degrees of freedom (a= 1.0 or larger).